Optimal. Leaf size=60 \[ \frac {2 i \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {a+b \sinh (x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2663, 2661} \[ \frac {2 i \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {a+b \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2661
Rule 2663
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx &=\frac {\sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{\sqrt {a+b \sinh (x)}}\\ &=\frac {2 i F\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{\sqrt {a+b \sinh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 60, normalized size = 1.00 \[ \frac {2 i \sqrt {\frac {a+b \sinh (x)}{a-i b}} F\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )}{\sqrt {a+b \sinh (x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {b \sinh \relax (x) + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sinh \relax (x) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 125, normalized size = 2.08 \[ -\frac {2 \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \relax (x )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \relax (x )\right ) b}{i b -a}}\, \EllipticF \left (\sqrt {-\frac {a +b \sinh \relax (x )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right )}{b \cosh \relax (x ) \sqrt {a +b \sinh \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b \sinh \relax (x) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a+b\,\mathrm {sinh}\relax (x)}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \sinh {\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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